# Interpretation of miscibility curves

I stumbled across miscibility curves in the section Solubility of Liquids in Liquids (Organic Chemistry, by Wallwork and Grant) today. I'm not sure if my problem is with the way the content was laid out in the book or whether I'm not fully understanding the material, but I feel the book has dealt with the topic in a rather, obscure way.

I've taken snapshots of it and uploaded a portion of my text below.

What I want to know is if I interpreted the text (the bit about miscibility curves) correctly, which I seriously doubt.

Here's my interpretation:

This is the miscibility curve for phenol in water blah-blah-blah. All the points present in the area bound by the curve with the x-axis (concentration axis) are points of immiscibility. The curve itself represents the threshold of miscibility.

For example, at 20°C, phenol is soluble in water up till the concentration corresponding to point A on the curve. However, for phenol concentrations corresponding to points between A and B (lying on the line AB), the phenol-water mixture is immiscible and the phenol and water form separate layers. Although, at concentrations corresponding to point B (and beyond), the phenol is completely soluble. When for a constant temperature, the miscibility curve maps to two concentrations on the x-axis (eg; points A and B in the graph), which represent thresholds of miscibility at that temperature, the two mixtures (i.e- at A and B) are called conjugate pairs.

I labored for over two hours to come to the above conclusion, but when I saw the portion of the text which mentions that the liquids A and B exist together (at equilibrium) for the given temperature, I had to throw in the towel.

I have a very strong feeling that a large part of my 'interpretation' is horribly incorrect. If anyone's got the time, could you explain what the text actually means?

Your interpretation seems correct. When you have a mixture with an overall average mass fraction of X, it separates into two fluid phases in equilibrium. One of these fluid phases has mass fraction A, and the other fluid phase has mass fraction B. You can determine the fraction of the total mass that is liquid of mass fraction A and liquid of mass fraction B as follows: let f represent the fraction that is liquid of mass fraction B, and (1-f) the fraction that is liquid of mass fraction A. Then $$m(1-f)x_A+mfx_B=mX$$where m is the total mass of both phases. Cancelling out the m's yields:$$(1-f)x_A+fx_B=X$$Solving for f yields:$$f=\frac{X-x_A}{x_B-x_A}$$This equation is called the "lever rule," because it operates like as see-saw. The fraction of the total amount of material that is in phase B is proportional to the difference between the average mass fraction and the mass fraction of your substance in phase A. And, the mass fraction of material that is in phase A is proportional to the difference between the average mass fraction and the mass fraction of your substance in phase B.